Arithmetic geometry of the moduli stack of Weierstrass fibrations over $\mathbb{P}^1$
Jun-Yong Park, Johannes Schmitt
TL;DR
This work constructs and analyzes the arithmetic geometry of moduli stacks for Weierstrass fibrations over unparameterized $ ext{P}^1$ by realizing them as quotients of Hom-stacks of maps from rational curves to weighted projective stacks. It develops a general stack-theoretic framework for maps from $ ext{P}^1$ to $ ext{P}(oldsymbol{ u})$, applies Romagny’s action theory to form quotient stacks, and uses GIT to obtain coarse moduli spaces and modular compactifications, with explicit stability criteria. The main geometric results show that the moduli stack $oldsymbol{ erd W_n}$ is smooth, and the minimal and stable substacks are separated Deligne–Mumford under suitable characteristics, while the arithmetic side yields a simple motivic description: for odd $n$, the motive of the stable Weierstrass fibrations stack is $oldsymbol{b L}^{10n-2}$, giving a weighted point count $oldsymbol{ abla q} = q^{10n-2}$. Together, these results connect the stack-theoretic construction with explicit motivic and finite-field invariants, and they provide a framework that extends to related moduli problems such as Silverman’s self-map spaces and level-structure variants of elliptic curves.
Abstract
Coarse moduli spaces of Weierstrass fibrations over the (unparameterized) projective line were constructed by the classical work of [Miranda] using Geometric Invariant Theory. In our paper, we extend this treatment by using results of [Romagny] regarding group actions on stacks to give an explicit construction of the moduli stack $\mathcal{W}_n$ of Weierstrass fibrations over an unparameterized $\mathbb{P}^{1}$ with discriminant degree $12n$ and a section. We show that it is a smooth algebraic stack and prove that for $n \geq 2$, the open substack $\mathcal{W}_{\mathrm{min},n}$ of minimal Weierstrass fibrations is a separated Deligne-Mumford stack over any base field $K$ with $\mathrm{char}(K) \neq 2,3$ and not dividing $n$. Arithmetically, for the moduli stack $\mathcal{W}_{\mathrm{sf},n}$ of stable Weierstrass fibrations, we determine its motive in the Grothendieck ring of stacks to be $\{\mathcal{W}_{\mathrm{sf},n}\} = \mathbb{L}^{10n - 2}$ in the case that $n$ is odd, which results in its weighted point count to be $\#_q(\mathcal{W}_{\mathrm{sf},n}) = q^{10n - 2}$ over $\mathbb{F}_q$. In the appendix, we show how our methods can be applied similarly to the classical work of [Silverman] on coarse moduli spaces of self-maps of the projective line, allowing us to construct the natural moduli stack and to compute its motive.